Spin wave theory. The spin Hamiltonian is transformed into a magnon Hamiltonian following the usual Holstein-Primakoff transformations: \(S_{m}^{+} = \left(2 S - b_{m}^{\dag } b_{m}^{}\right)^{1/2} b_m\) , \(S_{m}^{-} = b_{m}^{\dag } \left(2 S - b_{m}^{\dag } b_{m}^{}\right)^{1/2}\) , \(S_{m}^{z} = S-b_{m}^{\dag } b_{m}^{}\) . Only bilinear terms are retained. The Hamiltonian has the form \(\mathcal {H} =E_{0}-\sum _{mn} \left[\left(J_{mn} + i \nu _{mn} D_{mn}\right) S b_{m}^{\dagger } b_{n}^{} + \text{H.c.}\right] + h \sum _{m} b_{m}^{\dag } b_{m}^{}\) , where \(E_0\)  is ground state energy. \(\nu _{mn}=-\nu _{nm}\)  designates the direction of DM interaction. Subsequently, energy bands are obtained via diagonalization of the Hamiltonian matrix. The boundaries of the topological phase diagram was determined based on tracking gap closures and opening. [1]}, [2]}. The edge states were calculated for a strip sample in open boundary condition. In Fig. REF , we considered \(W=120\) , which is large enough to fix the edge states [1]}. Different edge conditions will give rise to different edge states. But, the topology features remain unchanged. Our selection of two edges is similar to Fig. REF . For further details on spin wave theory see Supplementary Information.
